\(\int (d+e x^2)^2 \sqrt {a+b \text {arccosh}(c x)} \, dx\) [552]

Optimal result

Integrand size = 22, antiderivative size = 672 \[ \int \left (d+e x^2\right )^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=d^2 x \sqrt {a+b \text {arccosh}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \text {arccosh}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \text {arccosh}(c x)}-\frac {\sqrt {b} d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {b} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {b} d e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {b} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {b} e^2 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {b} d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {b} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {b} d e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {b} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {b} e^2 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{320 c^5} \]

[Out]

-1/1600*e^2*exp(5*a/b)*erf(5^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*b^(1/2)*5^(1/2)*Pi^(1/2)/c^5-1/1600*e^2*e
rfi(5^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*b^(1/2)*5^(1/2)*Pi^(1/2)/c^5/exp(5*a/b)-1/72*d*e*exp(3*a/b)*erf(
3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*b^(1/2)*3^(1/2)*Pi^(1/2)/c^3-1/192*e^2*exp(3*a/b)*erf(3^(1/2)*(a+b*a
rccosh(c*x))^(1/2)/b^(1/2))*b^(1/2)*3^(1/2)*Pi^(1/2)/c^5-1/72*d*e*erfi(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2
))*b^(1/2)*3^(1/2)*Pi^(1/2)/c^3/exp(3*a/b)-1/192*e^2*erfi(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*b^(1/2)*3^
(1/2)*Pi^(1/2)/c^5/exp(3*a/b)-1/4*d^2*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c-1/8*d*
e*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c^3-1/32*e^2*exp(a/b)*erf((a+b*arccosh(c*x))
^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c^5-1/4*d^2*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c/exp(a/b
)-1/8*d*e*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c^3/exp(a/b)-1/32*e^2*erfi((a+b*arccosh(c*x)
)^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c^5/exp(a/b)+d^2*x*(a+b*arccosh(c*x))^(1/2)+2/3*d*e*x^3*(a+b*arccosh(c*x))^(
1/2)+1/5*e^2*x^5*(a+b*arccosh(c*x))^(1/2)

Rubi [A] (verified)

Time = 1.62 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.00, number of steps used = 42, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5909, 5879, 5953, 3388, 2211, 2236, 2235, 5884, 3393} \[ \int \left (d+e x^2\right )^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=-\frac {\sqrt {\pi } \sqrt {b} e^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {\frac {\pi }{5}} \sqrt {b} e^2 e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {\pi } \sqrt {b} e^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {\frac {\pi }{5}} \sqrt {b} e^2 e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {\pi } \sqrt {b} d e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} d e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {\pi } \sqrt {b} d e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} d e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {\pi } \sqrt {b} d^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {\pi } \sqrt {b} d^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}+d^2 x \sqrt {a+b \text {arccosh}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \text {arccosh}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \text {arccosh}(c x)} \]

[In]

Int[(d + e*x^2)^2*Sqrt[a + b*ArcCosh[c*x]],x]

[Out]

d^2*x*Sqrt[a + b*ArcCosh[c*x]] + (2*d*e*x^3*Sqrt[a + b*ArcCosh[c*x]])/3 + (e^2*x^5*Sqrt[a + b*ArcCosh[c*x]])/5
- (Sqrt[b]*d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*d*e*E^(a/b)*Sqrt[Pi]*
Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*c^3) - (Sqrt[b]*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sq
rt[b]])/(32*c^5) - (Sqrt[b]*d*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(24*c^
3) - (Sqrt[b]*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(64*c^5) - (Sqrt[b]*
e^2*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(320*c^5) - (Sqrt[b]*d^2*Sqrt[Pi]*
Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*c*E^(a/b)) - (Sqrt[b]*d*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sq
rt[b]])/(8*c^3*E^(a/b)) - (Sqrt[b]*e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(32*c^5*E^(a/b)) - (Sq
rt[b]*d*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(24*c^3*E^((3*a)/b)) - (Sqrt[b]*e^2*Sqr
t[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(64*c^5*E^((3*a)/b)) - (Sqrt[b]*e^2*Sqrt[Pi/5]*Erfi[
(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(320*c^5*E^((5*a)/b))

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 5879

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5884

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCosh[c*x])^n/(
m + 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])
), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 5909

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p
] && (p > 0 || IGtQ[n, 0])

Rule 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (d^2 \sqrt {a+b \text {arccosh}(c x)}+2 d e x^2 \sqrt {a+b \text {arccosh}(c x)}+e^2 x^4 \sqrt {a+b \text {arccosh}(c x)}\right ) \, dx \\ & = d^2 \int \sqrt {a+b \text {arccosh}(c x)} \, dx+(2 d e) \int x^2 \sqrt {a+b \text {arccosh}(c x)} \, dx+e^2 \int x^4 \sqrt {a+b \text {arccosh}(c x)} \, dx \\ & = d^2 x \sqrt {a+b \text {arccosh}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \text {arccosh}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \text {arccosh}(c x)}-\frac {1}{2} \left (b c d^2\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}} \, dx-\frac {1}{3} (b c d e) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}} \, dx-\frac {1}{10} \left (b c e^2\right ) \int \frac {x^5}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}} \, dx \\ & = d^2 x \sqrt {a+b \text {arccosh}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \text {arccosh}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \text {arccosh}(c x)}-\frac {d^2 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 c}-\frac {(d e) \text {Subst}\left (\int \frac {\cosh ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{3 c^3}-\frac {e^2 \text {Subst}\left (\int \frac {\cosh ^5\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{10 c^5} \\ & = d^2 x \sqrt {a+b \text {arccosh}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \text {arccosh}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \text {arccosh}(c x)}-\frac {d^2 \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 c}-\frac {d^2 \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 c}-\frac {(d e) \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}+\frac {3 \cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{3 c^3}-\frac {e^2 \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 \sqrt {x}}+\frac {5 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 \sqrt {x}}+\frac {5 \cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 \sqrt {x}}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{10 c^5} \\ & = d^2 x \sqrt {a+b \text {arccosh}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \text {arccosh}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \text {arccosh}(c x)}-\frac {d^2 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{2 c}-\frac {d^2 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{2 c}-\frac {(d e) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{12 c^3}-\frac {(d e) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 c^3}-\frac {e^2 \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{160 c^5}-\frac {e^2 \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{32 c^5}-\frac {e^2 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 c^5} \\ & = d^2 x \sqrt {a+b \text {arccosh}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \text {arccosh}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \text {arccosh}(c x)}-\frac {\sqrt {b} d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {(d e) \text {Subst}\left (\int \frac {e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{24 c^3}-\frac {(d e) \text {Subst}\left (\int \frac {e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{24 c^3}-\frac {(d e) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 c^3}-\frac {(d e) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 c^3}-\frac {e^2 \text {Subst}\left (\int \frac {e^{-i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{320 c^5}-\frac {e^2 \text {Subst}\left (\int \frac {e^{i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{320 c^5}-\frac {e^2 \text {Subst}\left (\int \frac {e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{64 c^5}-\frac {e^2 \text {Subst}\left (\int \frac {e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{64 c^5}-\frac {e^2 \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{32 c^5}-\frac {e^2 \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{32 c^5} \\ & = d^2 x \sqrt {a+b \text {arccosh}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \text {arccosh}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \text {arccosh}(c x)}-\frac {\sqrt {b} d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {(d e) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{12 c^3}-\frac {(d e) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{12 c^3}-\frac {(d e) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{4 c^3}-\frac {(d e) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{4 c^3}-\frac {e^2 \text {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{160 c^5}-\frac {e^2 \text {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{160 c^5}-\frac {e^2 \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{32 c^5}-\frac {e^2 \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{32 c^5}-\frac {e^2 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{16 c^5}-\frac {e^2 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{16 c^5} \\ & = d^2 x \sqrt {a+b \text {arccosh}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \text {arccosh}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \text {arccosh}(c x)}-\frac {\sqrt {b} d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {b} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {b} d e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {b} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {b} e^2 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{320 c^5}-\frac {\sqrt {b} d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {b} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 c^5}-\frac {\sqrt {b} d e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\sqrt {b} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 c^5}-\frac {\sqrt {b} e^2 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{320 c^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.91 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.80 \[ \int \left (d+e x^2\right )^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=\frac {b e^{-\frac {5 a}{b}} \left (450 e^{\frac {6 a}{b}} \left (8 a c^4 d^2 \sqrt {\frac {a}{b}+\text {arccosh}(c x)}+8 b c^4 d^2 \text {arccosh}(c x) \sqrt {\frac {a}{b}+\text {arccosh}(c x)}-b e \left (4 c^2 d+e\right ) \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \sqrt {-\frac {(a+b \text {arccosh}(c x))^2}{b^2}}\right ) \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c x)\right )-9 \sqrt {5} b e^2 \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \sqrt {-\frac {(a+b \text {arccosh}(c x))^2}{b^2}} \Gamma \left (\frac {3}{2},-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-e^{\frac {2 a}{b}} \left (25 \sqrt {3} b e \left (8 c^2 d+3 e\right ) \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \sqrt {-\frac {(a+b \text {arccosh}(c x))^2}{b^2}} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+450 e^{\frac {2 a}{b}} \left (8 a c^4 d^2 \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}}+8 b c^4 d^2 \text {arccosh}(c x) \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}}+b e \left (4 c^2 d+e\right ) \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \sqrt {-\frac {(a+b \text {arccosh}(c x))^2}{b^2}}\right ) \Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+b e e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \sqrt {-\frac {(a+b \text {arccosh}(c x))^2}{b^2}} \left (25 \sqrt {3} \left (8 c^2 d+3 e\right ) \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+9 \sqrt {5} e e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )\right )\right )}{7200 c^5 (a+b \text {arccosh}(c x))^{3/2}} \]

[In]

Integrate[(d + e*x^2)^2*Sqrt[a + b*ArcCosh[c*x]],x]

[Out]

(b*(450*E^((6*a)/b)*(8*a*c^4*d^2*Sqrt[a/b + ArcCosh[c*x]] + 8*b*c^4*d^2*ArcCosh[c*x]*Sqrt[a/b + ArcCosh[c*x]]
- b*e*(4*c^2*d + e)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)])*Gamma[3/2, a/b + ArcC
osh[c*x]] - 9*Sqrt[5]*b*e^2*Sqrt[a/b + ArcCosh[c*x]]*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)]*Gamma[3/2, (-5*(a + b
*ArcCosh[c*x]))/b] - E^((2*a)/b)*(25*Sqrt[3]*b*e*(8*c^2*d + 3*e)*Sqrt[a/b + ArcCosh[c*x]]*Sqrt[-((a + b*ArcCos
h[c*x])^2/b^2)]*Gamma[3/2, (-3*(a + b*ArcCosh[c*x]))/b] + 450*E^((2*a)/b)*(8*a*c^4*d^2*Sqrt[-((a + b*ArcCosh[c
*x])/b)] + 8*b*c^4*d^2*ArcCosh[c*x]*Sqrt[-((a + b*ArcCosh[c*x])/b)] + b*e*(4*c^2*d + e)*Sqrt[a/b + ArcCosh[c*x
]]*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)])*Gamma[3/2, -((a + b*ArcCosh[c*x])/b)] + b*e*E^((6*a)/b)*Sqrt[-((a + b*
ArcCosh[c*x])/b)]*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)]*(25*Sqrt[3]*(8*c^2*d + 3*e)*Gamma[3/2, (3*(a + b*ArcCosh
[c*x]))/b] + 9*Sqrt[5]*e*E^((2*a)/b)*Gamma[3/2, (5*(a + b*ArcCosh[c*x]))/b]))))/(7200*c^5*E^((5*a)/b)*(a + b*A
rcCosh[c*x])^(3/2))

Maple [F]

\[\int \left (e \,x^{2}+d \right )^{2} \sqrt {a +b \,\operatorname {arccosh}\left (c x \right )}d x\]

[In]

int((e*x^2+d)^2*(a+b*arccosh(c*x))^(1/2),x)

[Out]

int((e*x^2+d)^2*(a+b*arccosh(c*x))^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \left (d+e x^2\right )^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((e*x^2+d)^2*(a+b*arccosh(c*x))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \left (d+e x^2\right )^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=\int \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \left (d + e x^{2}\right )^{2}\, dx \]

[In]

integrate((e*x**2+d)**2*(a+b*acosh(c*x))**(1/2),x)

[Out]

Integral(sqrt(a + b*acosh(c*x))*(d + e*x**2)**2, x)

Maxima [F]

\[ \int \left (d+e x^2\right )^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=\int { {\left (e x^{2} + d\right )}^{2} \sqrt {b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

integrate((e*x^2+d)^2*(a+b*arccosh(c*x))^(1/2),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)^2*sqrt(b*arccosh(c*x) + a), x)

Giac [F]

\[ \int \left (d+e x^2\right )^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=\int { {\left (e x^{2} + d\right )}^{2} \sqrt {b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

integrate((e*x^2+d)^2*(a+b*arccosh(c*x))^(1/2),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^2*sqrt(b*arccosh(c*x) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \left (d+e x^2\right )^2 \sqrt {a+b \text {arccosh}(c x)} \, dx=\int \sqrt {a+b\,\mathrm {acosh}\left (c\,x\right )}\,{\left (e\,x^2+d\right )}^2 \,d x \]

[In]

int((a + b*acosh(c*x))^(1/2)*(d + e*x^2)^2,x)

[Out]

int((a + b*acosh(c*x))^(1/2)*(d + e*x^2)^2, x)